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It depends upon the individual mathematician's level of
sophistication whether he regards these difficulties as relevant
or of no consequence for the consistent development of mathe-
matics. The courageous analyst goes boldly ahead, piling one
Babel on top of another and trusting that no outraged god of
reason mil confound him and all his works, while the critical
logician, peering cynically at the foundations of his brother's
imposing skyscraper, makes a rapid mental calculation pre-
dicting the date of collapse. In the meantime all are busy and
all seem to be en joying themselves. But one conclusion appears
to be inescapable: without a consistent theory of the mathe-
matical infinite there is no theory of irrationals; without a
theory of irrationals there is no mathematical analysis hi any
form even remotely resembling what we now have; and finally,
without analysis the major part of mathematics - including
geometry and most of applied mathematics - as it now exists
would cease to exist.
The most important task confronting mathematicians would
therefore seem to be the construction of a satisfactory theory of
the infinite. Cantor attempted this, with what success wiB be
seen later. As for the DedeMnd theory of irrationals, its author
seems to have had some qualms, for he hesitated over two years
before venturing to publish it. If the reader will glance back at
Eudoxus* definition of 'same ratio' (Chapter 2) he will see that
"infinite difficulties* occur there too, specifically in the phrase
*any whatever equimultiples'. Nevertheless some progress has
been made since Eudoxus wrote; we are at least beginning to
understand the nature of our difficulties.
The other outstanding contribution which DedeMnd made
to the concept of 'number' was in the direction of algebraic
numbers. For the nature of the fundamental problem con-
cerned we must refer to what was said in the chapter OB
Kronecker concerning algebraic number fields and the resolu-
tion of algebraic integers into then: prime factors. The crux of
the matter is that in same such fields resolution into prime
factors is not unique as it is in common arithmetic; Dedekiad
restored this highly desirable uniqueness by the invention of
what he called ideals. An ideal is not a number, but an infinite